Single-channel color image encryption algorithm based on fractional Hartley

Single-channel color image encryption algorithm based

on fractional Hartley transform and vector operation

Ye Liu &Juan Du &Jinghui Fan &Lihua Gong

#Springer Science+Business Media New York 2013

Abstract A single-channel color image encryption algorithm is proposed by combining fractional Hartley transform (FRHT)with vector operation.The original color image is decomposed into RGB components and the G and B components are encrypted into two phase-only masks θG and θB with vector operation,respectively.The R ,θG and θB are transformed by FRHT and vector operation twice to obtain amplitude,random phase and decryption phase key.The new amplitude combined with the random phase is transformed by FRHT once more and then the result is scrambled by the chaotic scrambling to strengthen the security of the algorithm.The private phase key is dependent on the original image,which makes the proposed encryption algorithm more secure than the linear color image encryption algorithm based on the double random phase encoding in FRHT.Simulation results demon-strate the security and effectiveness of the proposed algorithm.

Keywords Vector operation .Fractional Hartley transform .Image encryption .

Information security

1Introduction

With the civilization and progress of the society,people pay more and more attention to the protection of personal privacy.Image is widely used as an effective information carrier because of its vivid and lively.In order to ensure the security of the private image information,image encryption is one of the effective approaches.Since Refregier and Javidi [13]first proposed the double random phase encoding in Fourier transform (FT)domain in 1995,it has been extended DOI

10.1007/s11042-013-1778-0

Y .Liu (*):J.Du :J.Fan :L.Gong

Department of Electronic Information Engineering,Nanchang University,Nanchang 330031,China e-mail:liuye@https://www.360docs.net/doc/eb2756613.html,

J.Du

e-mail:ncujdu@https://www.360docs.net/doc/eb2756613.html,

J.Fan

e-mail:jinghuifan@https://www.360docs.net/doc/eb2756613.html,

L.Gong

e-mail:ncuglh@https://www.360docs.net/doc/eb2756613.html,

L.Gong

Jiangxi Province Key Laboratory of Image Processing and Pattern Recognition,

Nanchang Hangkong University,Nanchang 330063,China

into Fresnel transform (FRT)domain [5,14],and fractional Fourier transform (FRFT)domain

[7,15,16].FRFT is the generalization of FT.The fractional orders of FRFT are extra keys to encrypt images compared with FT.Considering FT has its fractional form,researchers expect to find the fractional form for Hartley transform (HT).Some researchers define FRHT as the sum of the real and imaginary of the FRFT.The output of FRHT defined in this way for a real input is real,but it does not satisfy the additive property and has no inverse transform.In order to overcome the flaws,Pei and Ding [12]presented a redefined FRHT.Then some researchers encrypt an image by extending the double random encoding technique to the redefined FRHT

[6,18].But the redefined FRHT is not a real transform,Li and Zhao [8]suggested a special form named simplified fractional Hartley transform (SFRHT)for the redefined FRHT,and then proposed an image encryption method with SFRHT.Li and Zhao [9]also proposed a three-channel color image encryption algorithm using double random phase encoding based on FRHT in RGB space to extend the application field of the redefined FRHT to color image.Color image encryption has become an important part of image security not only for its beauty in vision but also for the meaningful practical use [4].Three-channel technique for color image encryption needs multiple light sources and sets of optical components which increase the difficulty and cost of the system [1,2,10],while single-channel technique only needs a light source and a set of optical components.Moreover the outcome of the latter is more compact and robust than that of the former [3,11].So single-channel technique is an important method and research direction for color image encryption.Considering FRHT is a linear transform and the linear encryption system based on double random phase encoding in FRHT domain is more vulnerable to some common attacks such as known-plaintext attack,chosen-plaintext attack,chosen-ciphertext attack and ciphertext-only attack [19,20],a new single channel color image encryption algorithm based on FRHT,vector operation and chaos theory is proposed.In this method,the private phase key θA 1is dependent on the original image.It is in vain to attack the encryption system for the private phase key to obtain any other original images decrypted by this system.

The remainder of this paper is organized as follows:Section 2represents some basic theories of the proposed method.The proposed algorithm is described in Section 3.Performance and security analyses are reported in Section 4.Conclusions are stated in Section 5.

2Basic theories

2.1Vector operation

The addition of two vectors [17]in the two-dimensional Cartesian coordinate system can be completed geometrically by the sum of two complex numbers.

z 3?z 1tz 2

e1TThe angle θbetween complex numbers z 1and z 2is

θ?π?acrcos z 1j j 2tz 2j j 2?z 3j j 2 =2z 1j j z 2j j eT e2T

so the addition of two unit vectors can be accomplished geometrically in the polar coordinate system by

z 3?exp i ?1eTtexp i ?2eTe3T

where ?1and ?2are the arguments of complex numbers z 1and z 2,respectively.It is easy to get

z 3j j 2?exp i ?1eTtexp i ?2eTj j 2e4T

Consequently,|z 3|2can be decomposed into two phases ?1and ?2.The pixel values of a normalized image are between 0and 1,so are the pixel square root values.It is obvious that the pixel square root values of a normalized image and the two unit vectors satisfy the triangle inequality.Therefore,a pixel value f of a normalized grayscale image can be regarded as the square of the modulus of two complex numbers whose modulus values equal 1and represented as

f ?exp i ?eTtexp i ?tθeT? j j 2e5T

where ?is an arbitrary random phase whose values are between 0and 2π,θis between 0and πand satisfies

θ?π?arccos 1?f =2eTe6T

Equations (6)and (7)are the forward and inverse transforms of the vector operation.As a result,a normalized grayscale image can be encoded into a phase-only mask θwithout any iterative process.

2.2Fractional Hartley transform

As for an input grayscale image f (x ,y ),the two dimensional FRHT H [6,9,18]can be defined as an addition of two FRFTs.

H p 1;p 2f x ;y eT? x 0;y 0

eT?1te i ?1t?2eT=22F p 1;p 2f x ;y eT? x 0;y 0eTt1?e i ?1t?2eT=22F p 1t2;p 2t2f x ;y eT? x 0;y 0eTe7T

where p 1and p 2are fractional orders,?1=p 1π/2and ?2=p 2π/2,and F is the FRFT.

From Eq.(7)and the properties of FRFT,one can obtain that the FRHT satisfies the index additive property,and has the period with 2.Moreover,when p 1=p 2=1,the FRHT reduces to HT.

The inverse transform of the FRHT is the transform H with the opposite fractional orders ?p 1and ?p 2.

3Single-channel color image encryption based on FRHT and vector operation

3.1Encryption process

For a color image with M ×N ×3pixels,the proposed encryption process is shown in Fig.1.The detail steps are described as follows:

(1)The normalized G and B components of the original color image are respectively

converted to phase-only masks θG and θB with vector operation.

θG ?π?arccos 1?G =2eT

e8TθB ?π?arccos 1?B =2eTe9T

(2)The R component combined with phase θG is encoded into complex amplitude,and

subsequently transformed by FRHT.

A 0exp i ?0eT?H a R exp i θG eT? e10T

Then the amplitude A 0and the phase ?0can be extracted from the result and then converted into phase-only mask θA 020;π? and real amplitude A ?0∈0;255? by

θA 0?π?arccos 1?A 0=2eT

e11TA ?0??0=2πeT?255e12T

(3)A ?0and θB are regarded as the amplitude and phase of the input data respectively and

then transformed by repeating step (2).The output is

θA 1?π?arccos 1?A 1=2eT

e13TA ?1??1=2πeT?255e14T

where the θA 1is recorded and transmitted as the private phase key of decryption.

(4)To strengthen the security of the encryption system,FRHT is performed once more

A 2exp i ?2eT?H a A ?1exp i θA 0eT??e15T

(5)The above steps just change the value of the pixel.To further change the position of the

pixel,A 2is permutated by the chaotic scrambling algorithm in Ref.[21]to obtain the final encrypted image C .The phase ?2is its accompanying phase.

To decrease the number of keys,bifurcation parameter u and the number of discarding values K of the logistical map in the chaotic scrambling process can be set as constants.Fig.1Schematic of encryption

3.2Decryption process

With the ciphertext C,its accompanying phase?2and the keys including the fractional

order a of FRHT,the seed of the chaotic scrambling s and the private phase keyθA

1 generated in the encryption process,the decryption steps is just the inverse process of the

encryption steps.The final decrypted color image can be obtained by combining the

RGB components together.

4Numerical simulations and discussions

The color image‘Brain’with the size of290×290shown in Fig.2a is used as the test

image.The order a of the FRHT is set as0.5.The seed s of the chaotic scrambling is set

as0.221.The amplitude and the phase distributions of the encrypted image are shown in

Fig.2b and c respectively.The decrypted color image with all correct keys is shown in

Fig.2d.

Figure3shows the decrypted color image with incorrect keys.Figure3a shows the

decrypted color image with the incorrect FRHT order a=0.5+0.01while the other keys are

correct.Figure3b shows the decrypted color image with incorrect seed s=0.221+1×10?15

while the other keys are correct.Figure3c displays the incorrect private phase keyθA

1 interfered by a random phase produced by computer while the other keys are correct.When

the deviation of the fractional order is larger than0.01,the deviation of the seed is larger than

1×10?15or the private decryption phase key is incorrect,the decrypted image cannot be

recognized.

4.1Statistical analysis

Statistical analysis of the proposed image encryption algorithm can be tested from two aspects.

One is to test the histograms of the encrypted images of different color images and the other is

to test the correlations of adjacent pixels of the plain image and its corresponding encrypted

image.Figure4a and b display the histograms of the encrypted images of two different original

color images‘brain’and‘Lena’respectively.

Seeing from Fig.4,when the proposed encryption algorithm is carried out on different

original color images brain with290×290pixels and Lena with512×512pixels,the distri-

butions of the histograms of the encrypted images are similar.Therefore,the attackers cannot

obtain any valid information through statistical analysis.

Fig.2a Original color image‘brain’,b amplitude distribution of the encrypted image,c phase distribution of the encrypted image and d decrypted image with correct keys

The correlation coefficient r xy of two adjacent pixels of a grayscale image can be calculated by

r xy ?cov x ;y eT??????????D x eTp ???????????D y eT

p e16Tcov x ;y eT?1X i ?1K

x i ?E x eTeTy i ?E y eTeT? e17Twhere x and y are the gray levels of two adjacent pixels in the image,E x eT?1∑i K

x i and

D x eT?1K

∑i ?1K x i ?E x eT? 2

.To test the correlation between two adjacent vertical,horizontal or diagonal pixels in the original or encrypted images,5,000pairs of adjacent pixels of the image are chosen randomly.Table 1displays the correlation coefficients of the original image and the encrypted image.The correlation coefficient of the plaintext is enormous in each direction of each component,while the correlation coefficient of the encrypted image is tiny in each direction.The adjacent pixels of the encrypted image are almost no relationship.Therefore,the encryption algorithm removes the tight relationship between adjacent pixels of the plaintext successfully.The results indicate that the proposed algorithm improves the ability of the encryption algorithm to resist statistical

analysis.Fig.3Decrypted color images with:a incorrect order a ,b incorrect seed s ,c incorrect private phase θA

1a

b 0500100015002000

25003000

0501001502002500100

200

300

400

500

600

700

800

9001000050100150200250Fig.4a Histogram of ciphertext of Lena and b histogram of ciphertext of Brain

4.2Sensitivity of the keys

To express the quality of the decrypted image,the mean square error (MSE)between the original and the decrypted grayscale image is introduced as

MSE ?1M ?N

X N i ;j ?1f 0i ;j eT?f i ;j eTj j 2e18Twhere M and N separately represent the height and width of the image,f ’(i ,j )and f (i ,j )denote the pixel values of the decrypted image and the original image,respectively.The MSE of the color image is

MSE ?MSE r tMSE g tMSE b

3e19T

where MSE r ,MSE g ,MSE b are computed by using Eq.(18).

Figure 5a and b show the deviation of MSE versus fractional order a and the seed of the chaotic scrambling s ,severally.The MSE approximates to zero when the image is decrypted with correct fractional order or seed,whereas the MSE increases sharply when the fractional order or seed slightly departs from the correct value.The order a and the seed s are both very sensitive seeing from Figs.3and 5.

Table 1Correlation of two adjacent pixels of the original and encrypted image

Correlation coefficient Original image

Encrypted image R

G B Encrypted amplitude Encrypted phase Horizontal

0.98760.97970.95160.00840.0247V ertical

0.98740.97990.95160.00870.0472Diagonal 0.98770.97710.94720.00450.0299

a b

1

2

34d M S E x 10-14d

M S E Fig.5MSE versus the deviation of (a )the fractional order a and (b )the seed s

4.3Key space analysis

Since the chaotic scrambling algorithm is independent from the FRHT,the key space of the cryptosystem consists of FRHT keys and chaotic keys.As the bifurcation parameter u and the number of discarding values K in the chaotic scrambling process are set as constants,to calculate the key space of the chaotic scrambling module denoted by S 1,we only need to consider the key space of the seed s .The key space of the seed S 1is 1015from Fig.3.In the

process of FRHT,only the key space of the private phase key θA 10is considered and denoted

by S 2.In order to evaluate the key space of the private phase key,we consider the phase key

θA 1is fluctuating in certain range,and the pseudo-key θA 10close to correct key is expressed as

θA 10?θA 1td ΔθA 1e20T

where ΔθA 1is a random phase function whose value is located in (0,2π),θA 1is the correct key and d is a coefficient whose value is located in (?1,1).Figure 6shows the deviation of MSE versus the perturbation of private phase key θA 1.Here,5,000is regarded as the threshold.When the MSE is more than 5,000,we cannot obtain any valid information from the decrypted image in vision.The maximum Δd is 0.1241.The key space of the private phase S 2is huge as (2π/0.1241)290×290?51290×290.So the whole key space of the cryptosystem is huge as S 1×S 2=51290×290×1015.Therefore the key space of the proposed algorithm is enormous enough to resist brute-force attack.

4.4Noise attack

The ciphertext affected by noise can be expressed as

C 0?C tkG e21T

where C ’and C are the noisy ciphertext and the ideal ciphertext,respectively.k is the coefficient of the noise strength.G is the Gaussian random noise with zero-mean and identity standard deviation.Figure 7shows the decrypted images when k equals to 1,10,20,

d

M S E Fig.6MSE versus the perturbation of the private phase key θA 1

respectively.With the k increasing,the recovered image becomes more and more ambiguous.From the results we can realize that even when k =20,the main information of the image can also be recognized.So the proposed technique is secure against noise attack.

4.5Comparison of our approach with existing method

Common attacks such as known-plaintext attack,chosen-plaintext attack and chosen-ciphertext attack require plaintext –ciphertext pairs.Chosen-plaintext/ciphertext attack even requires specific plaintexts/ciphertexts.In a linear encryption system,it is obvious that the ciphertext of αP 1+βP 2is αC 1+βC 2where P 1,P 2are original images,C 1,C 2are their corresponding encrypted images and α,βare two scalar coefficients.So a cryptanalyst can use linear equations C =TP or delta functions formed by particular linear combinations of P 1,P 2or C 1,C 2to attack the encryption system.Our proposed algorithm is more secure against the above attacks compared with the linear color image encryption algorithm proposed in [9],not only because our method is nonlinear with the use of nonlinear transform namely vector operation,but also because the two random phases used in [9]are independent of the original image while the private phase key θA 1generated in the encryption process is dependent on the original image.That is to say,in the former method,the wanted original images can be recovered by the phase keys reconstructed by the above attacks,while in the latter method,the wanted original images cannot.

5Conclusion

By applying the FRHT,vector operation and chaos theory,a single-channel color image encryption algorithm is https://www.360docs.net/doc/eb2756613.html,paring with HT,FRHT can provide fractional orders as the additional keys to improve the security of the algorithm,but it is linear.In order to strengthen the security of the system,vector operation and logistic map are further applied to make the image encryption system nonlinear and disorder.The vector operation is simple and fast since it can encrypt a grayscale image into a phase-only mask without any iterative process.Moreover the proposed scheme is performed in a single channel which is more compact and robust than conventional multiple-channel encryption system.The original color image cannot be recovered unless all of the keys including fractional order,the seed of logistic map and private phase are all correct.Moreover the private phase key relies on the original color image.It is impossible to obtain any information of other original images by

the

Fig.7Results of noise attack:a k =1,b k =10,c k =20

accompanying private phase key of the certain original image.Simulation results demonstrate that the proposed algorithm is secure with sensitive keys,large key space,anti-noise ability and anti-statistic ability.

Acknowledgments The work is supported by the National Natural Science Foundation of China(Grant Nos. 61262084and61141007),the Foundation for Young Scientists of Jiangxi Province(Jinggang Star)(Grant No. 20122BCB23002),and the Opening Project of Key Laboratory of Image Processing and Pattern Recognition (Nanchang Hangkong University),Jiangxi Province(Grant No.TX201204002).

References

1.Chen W,Chen XD(2011)Optical color image encryption based on an asymmetric cryptosystem in the

Fresnel domain.Opt Commun284:3913–3917

2.Chen LF,Zhao DM(2009)Color image encoding in dual fractional Fourier-wavelet domain with random

phases.Opt Commun282:3433–3438

3.Deng XP,Zhao DM(2012)Single-channel color image encryption based on asymmetric cryptosystem.Opt

Laser Technol44:136–140

4.Guo Q,Liu ZJ,Liu ST(2010)Color image encryption by using Arnold and discrete fractional random

transforms in IHS space.Opt Laser Eng48:800–805

5.Hwang HE,Han P(2006)Fast algorithm of phase mask for image encryption in Fresnel domain.J Opt Soc

Am A23:1870–1874

6.Jimenez C,Torres C,Mattos L(2011)Fractional Hartley transform applied to optical image encryption.J

Phys.doi:10.1088/1742-6596/274/1/012041

7.Joshi M,Shakher C,Singh K(2007)Color image encryption and decryption using fractional Fourier

transform.Opt Commun279:35–42

8.LI XX,Zhao DM(2008)Optical Image encryption with simplified fractional Hartley transform.Chinese

Phys Lett25:2477–2480

9.Li XX,Zhao DM(2010)Optical color image encryption with redefined fractional Hartley transform.Optik

121:673–67

10.Liu ZJ,Da JM,Sun XG,Liu ST(2010)Color image encryption by using the rotation of color vector in

Hartley transform domains.Opt Laser Eng48:800–805

11.Madhusudan J,Chandra S,Kehar S(2008)Color image encryption and decryption for twin images in

fractional Fourier domain.Opt Commun281:5713–15720

12.Pei SC,Ding JJ(2002)Fractional cosine,sine,and Hartley transforms.IEEE Trans Signal Process50:1661–

1680

13.Refregier P,Javidi B(1995)Optical image encryption based on input plane and Fourier plane random

encoding.Opt Lett20:767–769

14.Situ G,Zhang J(2004)Double random-phase encoding in the Fresnel domain.Opt Lett29:1584–1586

15.Tao R,Xin Y,Wang Y(2007)Double image encryption based on random phase encoding in fractional

Fourier domain.Opt Express15:16067–16079

16.Unnikrishnan G,Joseph J,Singh K(2000)Optical encryption by double-random phase encoding in the

fractional Fourier domain.Opt Lett25:887–889

17.Wang XG,Zhao DM(2011)Image encoding based on coherent superposition and basic vector operations.

Opt Commun284:945–951

18.Zhao DM,Li XX,Chen LF(2008)Optical image encryption with redefined fractional Hartley transform.

Opt Commun281:5326–5329

19.Zhou NR,Wang YX,Gong LH(2011)Novel optical image encryption scheme based on fractional Mellin

transform.Opt Commun284:3234–3242

20.Zhou NR,Wang YX,Gong LH,Chen XB,Yang YX(2012)Novel color image encryption algorithm based

on the reality preserving fractional Mellin transform.Opt Laser Technol44:2270–2281

21.Zhou NR,Wang YX,Gong LH,He H,Wu JH(2011)Novel single-channel color image encryption

algorithm based on chaos and fractional Fourier transform.Opt Commun284:2789–2796

Y e Liu received the B.S.degree in Radio Engineering from Hefei University of Technology,Hefei,China,in 1982.She is currently a professor of the electronic information engineering at Nanchang University.Her areas of interests are media security and wireless sensor network.Email:liuye@https://www.360docs.net/doc/eb2756613.html,

Juan Du received the B.S.degree in Communication Engineering from Nanchang University,China,in2011. She is currently pursuing the Master Degree in Signal and Information Processing at Nanchang University.Her areas of interests are multimedia security and image processing.

Jinghui Fan received his Bachelor Degree and Master Degree from Nanchang University in2001and2004, respectively.He is currently a lecturer of Department of Electronic Information Engineering,Nanchang Univer-sity.His interests are signal processing and information security.Email:jinghuifan@https://www.360docs.net/doc/eb2756613.html,

Lihua Gong received her Master Degree in Engineering in Electronic&Communication Engineering from Nanchang University in2011and Bachelor Degree in Physics from Jiangxi Normal University in2001.Since 2006she served as one of the Faculty of Department of Electronic Information Engineering,Nanchang University.She has published over20papers on information security in refereed international conferences and

journals.